Some cubic modular identities of Ramanujan

Title: Some cubic modular identities of Ramanujan
Creator: Borwein, J. M.; Borwein, P. B.; Garvan, F. G.
Relation: Transactions of the American Mathematical Society Vol. 343, Issue 1, p. 35-47
Publisher Link: http://dx.doi.org/10.1090/S0002-9947-1994-1243610-6
Publisher: American Mathematical Society (AMS)
Resource Type: journal article
Date: 1994
Description: There is a beautiful cubic analogue of Jacobi's fundamental theta function identity: θ⁴₃ = θ⁴₄ + θ⁴₂. It is ([formula cannot be replicated] q^n²+nm+m²)³ = ([formula cannot be replicated] ω^n-mq^n²+nm+m²)³ + ([formula cannot be replicated] q^{(n+1/3)²+(n+1/3)(m+1/3)+(m+1/3)²})³. Here ω = exp(2π i/3). In this note we provide an elementary proof of this identity and of a related identity due to Ramanujan. We also indicate how to discover and prove such identities symbolically.
Subject: theta functions; q-series; eta function; modular forms; cubic modular equations; hypergeometric functions
Identifier: http://hdl.handle.net/1959.13/940448
Identifier: uon:13009
Identifier: ISSN:0002-9497
Rights: First published in Transactions of the American Mathematical Society in Vol. 343, No. 1, pp. 35-47, 1994, published by the American Mathematical Society.
Language: eng
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